Dispersive Vertex Guarding for Simple and Non-Simple Polygons

TL;DR

It is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2, and an algorithm is provided that places vertex guards in simple polygons at dispersion distance at least $2.

Abstract

We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon $\mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at least $1$, and a rational number $\ell$; decide whether there is a set of vertex guards such that $\mathcal{P}$ is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least $\ell$. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2$. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least $2$. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most $2$.

Figures (8)

• Figure 1: A polygon in which edges have similar length.
• Figure 2: A polygon for which the optimal guard numbers for AGP and Dispersive AGP differ considerably.
• Figure 3: A variable gadget.
• Figure 4: A clause gadget.
• Figure 5: A split gadget.

Theorems & Definitions (6)

• Theorem 1
• Lemma 2
• Lemma 4
• Theorem 5
• Lemma 9
• Theorem 10